Theorem prnz 4734

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4719	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4296	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440  ∅c0 4285  {cpr 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3442  df-dif 3904  df-un 3906  df-nul 4286  df-sn 4581  df-pr 4583

This theorem is referenced by:  opnz  5421  propssopi  5456  fiint  9227  wilthlem2  27035  upgrbi  29166  wlkvtxiedg  29698  shincli  31437  chincli  31535  constrextdg2lem  33905  spr0nelg  47718  sprvalpwn0  47725